Fine regularity of L\'evy processes and linear (multi)fractional stable motion

In this work, we investigate the fine regularity of L\'evy processes using the 2-microlocal formalism. This framework allows us to refine the multifractal spectrum determined by Jaffard and, in addition, study the oscillating singularities of L\'evy processes. The fractal structure of the latter is proved to be more complex than the classic multifractal spectrum and is determined in the case of alpha-stable processes. As a consequence of these fine results and the properties of the 2-microlocal frontier, we are also able to completely characterise the multifractal nature of the linear fractional stable motion (extension of fractional Brownian motion to {\alpha}-stable measures) in the case of continuous and unbounded sample paths as well. The regularity of its multifractional extension is also presented, indirectly providing an example of a stochastic process with a non-homogeneous and random multifractal spectrum.


Introduction
The study of sample path continuity and Hölder regularity of stochastic processes is a very active field of research in probability theory. The existing literature provides a variety of uniform results on local regularity, especially on the modulus of continuity, for rather general classes of random fields (see e.g. Marcus and Rosen [30], Adler and Taylor [2] on Gaussian processes and Xiao [45] for more recent developments).
On the other hand, the structure of pointwise regularity is often more complex, in particular as it often happens to behave erratically as time passes. Such sample path behaviour was first put into light for Brownian motion by Orey and Taylor [33] and Perkins [34]. They respectively studied fast and slow points, which characterize logarithmic variations of the pointwise modulus of continuity, and proved that the sets of times with a given pointwise regularity have a particular fractal geometry. Khoshnevisan and Shi [25] recently extended the fast points study to fractional Brownian motion.
As exhibited by Jaffard [22], Lévy processes with a jump compound also display an interesting pointwise behaviour. Indeed, for this class of random fields, the pointwise exponent varies randomly inside a closed interval as time passes. This seminal work has been enhanced and extended by Durand [18], Durand and Jaffard [19] and Barral et al. [10]. Particularly, the latter proves that Markov processes have a range of admissible pointwise behaviours much wider than Lévy processes. In the aforementioned works, multifractal analysis happens to be the key concept for the study and the characterisation of local fluctuations of pointwise Multifractal analysis is concerned with the study of the level sets of the pointwise exponent, usually called the iso-Hölder sets of f , To describe the geometry of the collection (E h ) h∈R+ , and thereby to determine the arrangement of the Hölder regularity, we are interested in the local spectrum of singularities of f . It is usually denoted d f (h, V ) and defined by where O denotes the collection of nonempty open sets of R and dim H the Hausdorff dimension (by convention dim H (∅) = −∞).
Although (E h ) h∈R+ are random sets, stochastic processes such as Lévy processes [22], Lévy processes in multifractal time [9] and fractional Brownian motion happen to have a deterministic multifractal spectrum. Furthermore, these random fields are also said to be homogeneous as the Hausdorff dimension d X (h, V ) is independent of the set V for all h ∈ R + . When the pointwise exponent is constant along sample paths, the spectrum is described as degenerate, i.e. its support is reduced to a single point (e.g. the Hurst exponent in the case of f.B.m.). Nevertheless, Barral et al. [10] and Durand [17] provided examples of respectively Markov jump processes and wavelet random series with non-homogeneous and random spectrum of singularities.
As stated in Equations (1.1) and (1.2), classic multifractal analysis deals with the study of the variations of pointwise regularity. Unfortunately, it is known that common Hölder exponents (local and pointwise as well) do not give a complete picture of the local regularity (see e.g the deterministic Chirp function t → |t| α sin |t| −β detailed in [20]). Furthermore, they also lack of stability under the action of pseudo-differential operators.
2-microlocal analysis is one natural way to overcome these issues and obtain a more precise description of the local regularity. It has first been introduced by Bony [13] in the deterministic frame to study properties of generalized solutions of PDE. More recently, Herbin and Lévy Véhel [20] and Balança and Herbin [7] developed a stochastic approach based on this framework to investigate the finer regularity of stochastic processes such as Gaussian processes, martingales and stochastic integrals. In order to the study sample path properties in this frame, we need to recall the concept of 2-microlocal space. Definition 2. Let t ∈ R, s ≤ 0 and σ ∈ (0, 1) such that σ −s / ∈ N. A function f : R → R d belongs to the 2-microlocal space C σ,s t if there exist C > 0, ρ > 0 and a polynomial P t such that for all u, v ∈ B(t, ρ).
The time-domain characterisation (1.3) of 2-microlocal spaces has been obtained by Seuret and Lévy Véhel [41]. The original definition given by Bony [13] relies on the Littlewood-Paley decomposition of tempered distributions, and thereby corresponds to a description in the Fourier space. Another characterisation based on wavelet expansion has also been exhibited by Jaffard [21]. The extension of Definition 2 to σ / ∈ (0, 1) relies on the following important property satisfied by 2-microlocal spaces (see Theorem 1.1 in [23]), (1. 4) where I α f designates the fractional integral of f of order α. As a consequence of (1.4), the application of Equation (1.3) to iterated integrals or differentials of f provides an extension of Definition 2 to any σ ∈ R \ Z, which is sufficient for the purpose of this paper.
Similarly to the pointwise Hölder exponent, the introduction of 2-microlocal spaces leads naturaly to the definition of regularity tool named the 2-microlocal frontier and given by 2-microlocal spaces enjoy several inclusion properties which imply that the map s → σ f,t (s ) is well-defined and display the following features: • σ f,t (·) is a concave non-decreasing function; • σ f,t (·) has left and right derivatives between 0 and 1.
As a function, the 2-microlocal frontier σ f,t (·) offers a more complete description of the local regularity. In particular, it embraces the local Hölder exponent since α f,t = σ f,t (0). Furthermore, as stated in [32], if the modulus of continuity of f satisfies ω(h) = O (1/|log(h)|), the pointwise exponent can also be retrieved using the formula α f,t = − inf{s : σ f,t (s ) ≥ 0}. Note that the previous formula can not directly deduced from Equation 1.3 since Definition 2 does not stand when σ = 0. [32] provides an example of generalized function which does not satisfy this relation.
As observed [20], Brownian motion provides a simple instance of 2-microlocal frontier in the stochastic frame: almost surely for all t ∈ R, σ B,t satisfies In this paper, the 2-microlocal approach is combined with the classic use of multifractal analysis to obtain a finer description of the regularity of stochastic processes. Following the path of [22,18,19], we refine the multifractal description of Lévy processes (Section 2) and observe in particular that the use of the 2-microlocal formalism allows to capture subtle behaviours that can not be characterized by the classic spectrum of singularities.
This finer analysis of sample path properties of Lévy processes happens to be very useful for the study of another class of processes named linear fractional stable motion (LFSM). The LFSM is a common α-stable self-similar process with stationary increments, and can be seen as an extension of the fractional Brownian motion to the non-Gaussian frame. Since it also has long range dependence and heavy tails, it is of great interest in modelling. In Section 3, we completely characterize the multifractal nature of the LFSM, and thereby illustrate the fact that 2-microlocal analysis is well-suited to study the regularity of unbounded sample paths as well as continuous ones.

Statement of the main results
As it is well known, an R d -valued Lévy process (X t ) t∈R+ has stationary and independent increments. Its law is determined by the Lévy-Khintchine formula (see e.g. [40]): for all t ∈ R + and λ ∈ R d , E[e i λ,Xt ] = e tψ(λ) where ψ is given by Q is a non-negative symmetric matrix and π a Lévy measure, i.e. a positive Radon measure Throughout this paper, it will always be assumed that π(R d ) = +∞. Otherwise, the Lévy process simply corresponds to the sum of a compound Poisson process with drift and a Brownian motion whose regularity can be simply deduced.
Sample path properties of Lévy processes are known to depend on the growth of the Lévy measure near the origin. More precisely, Blumenthal and Getoor [12] defined the following exponents β and β , Owing to π's definition, β, β ∈ [0, 2]. Pruitt [36] proved that α X,0 a.s. = 1/β when Q = 0. Note that several other exponents have been defined in the literature focusing on Lévy processes sample paths properties (see e.g. [26,27] for some recent developments).
Jaffard [22] studied the spectrum of singularities of Lévy processes under the following assumption on the measure π, (1.7) Under the Hypothesis (1.7), Theorem 1 in [22] states that the multifractal spectrum of a Lévy process X is almost surely equal to (1.8) Note that Equation (1.8) still holds when β = 0. Durand [18] extended this result to Hausdorff g-measures, where g is a gauge function, and Durand and Jaffard [19] generalized the study to multivariate Lévy fields.
We establish in Proposition 1 a new proof of the multifractal spectrum (1.8) which does not require Assumption (1.7). We observe that results obtained in [18] on Hausdorff gmeasure are also extended using this method.
In order to refine the spectrum of singularities (1.8), we focus on the study of the 2-microlocal frontier of Lévy processes. For that purpose, we introduce and study the collections of sets ( E h ) h∈R+ and ( E h ) h∈R+ respectively defined by The family ( E h ) h∈R+ represents the set of times at which the 2-microlocal behaviour is rather common (and thus similar the 2-microlocal frontier (1.5) of Brownian motion), whereas at points which belong ( E h ) h∈R+ , the 2-microlocal frontier has an unusual form, with in particular a slope different from 1 at s = −h. The next statement gathers our main result on the 2-microlocal regularity of Lévy processes. Theorem 1. Sample paths of a Lévy process X almost surely satisfy (1.9) The collection of sets ( E h ) h∈R+ enjoys almost surely

Remark 1. The previous statement induces that dim
Hence, from a Hausdorff dimension point of view, the majority of the times t ∈ R + have a rather classic 2-microlocal frontier s → (α X,t + s ) ∧ 0.

Remark 2.
The collection of sets ( E h ) h∈R+ illustrates the fact that 2-microlocal analysis can capture particular behaviours that are not necessarily described by a classic multifractal spectrum.
Examples 1 and 2 constructed in Section 2.3 show that different behaviours may occur, depending on properties of the Lévy measure. The first one provides a class of Lévy processes which satisfy E h = ∅ for all h ∈ [0, 1/β ]. On the other hand, in Example 2 is constructed a collection of Lévy measures (π h ) h∈(1/2β ,1/β ) such that the related Lévy process almost surely enjoys E h = ∅.
It remains an open question to completely characterize the collection ( E h ) h∈R+ in terms of the Lévy measure π (Examples 1 and 2 indeed prove that the Blumenthal-Getoor exponent β is not sufficient).

Remark 3.
Although sample paths of Lévy processes do not satisfy the condition ω(h) = O(1/|log(h)|) outlined in the introduction, Theorem 1 nevertheless ensures that the pointwise Hölder exponent can be retrieved from the 2-microlocal frontier at any t ∈ R + using the formula α X,t = − inf{s : σ X,t (s ) ≥ 0}.
Since this work extends the study of the classic spectrum of singularities, it is also quite natural to investigate geometrical properties of the sets (E σ,s ) σ,s ∈R defined by Theorem 1 induces the next statement.
Corollary 1 generalizes the multifractal formula (1.8) since the spectrum of singularities corresponds to the case σ = 0. Note that the subtle behaviour exhibited in Theorem 1 is not captured by Equality (1.11). As outlined in the proof (Section 2.2), this property disappears because the sets ( E h ) h are negligible compared to ( E h ) h in terms of Hausdorff dimension.
Regularity results established in Theorem 1 also happen to be interesting outside the scope of Lévy processes, thanks to the powerful properties satisfied the 2-microlocal frontier. More precisely, it allows to characterized the multifractal nature of the linear fractional stable motion (LFSM). This process is usually defined by the following stochastic integral (see e.g. [39]) where M α,β is an α-stable random measure and H ∈ (0, 1). Several regularity properties have been determined in the literature. In particular, sample paths are known to be nowhere bounded [29] if H < 1/α, whereas they are Hölder continuous when H > 1/α. In this latter case, Takashima [44], Kôno and Maejima [28] proved that the pointwise and local Hölder exponents satisfy almost surely H − 1/α ≤ α X,t ≤ H and α X,t = H − 1/α. In the sequel, we will assume that α ∈ [1, 2), which is in particular required to obtain Hölder continuous sample paths (H > 1/α).
Using an alternative representation of LFSM obtained in Proposition 2, we enhance the aforementioned regularity results and obtain a description of the multifractal spectrum of the LFSM. Theorem 2. Let X be a linear fractional stable motion parametrized by α ∈ [1, 2) and H ∈ (0, 1). It satisfies almost surely for all In the continuous case H > 1/α, Theorem 2 ensures that the multifractal spectrum (σ = 0) of the LFSM is equal to (1.14) Spectrum 1.14 and Equation (1.13) clearly extend the aforementioned lower and upper bounds obtained on the pointwise and local Hölder exponents. We also note that as it could be expected, the LFSM is an homogeneous multifractal process.

Remark 5.
More generally, we observe that Theorem 2 unifies in terms of regularity the continuous (H > 1 α ) and unbounded (H < 1 α ) cases. Indeed, in both situations, the domain of acceptable 2-microlocal frontiers have the same multifractal structure. When H > 1 α , it intersects the s -axis, which induces α X,t > 0 and therefore the continuity of trajectories owing to properties of the 2-microlocal frontier.
On the contrary, when the domain is located strictly below the s -axis, it implies that sample paths are nowhere bounded. Nevertheless, the proof of Theorem 2 ensures in this case the existence of modification of the LFSM such that sample paths are tempered distributions whose 2-microlocal regularity can be studied as well. Figure 1 illustrates this dichotomy. Figure 1: Domains of admissible 2-microlocal frontiers for the LFSM An equivalent result is obtained in Proposition 3 for a similar class of processes called fractional Lévy processes (see [11,31,15]).
The LFSM admits a natural multifractional extension which has been introduced and studied in [42,43,16]. The definition of the linear multifractional stable motion (LMSM) is given by equation (1.12) where the Hurst exponent H is replaced by a function t → H(t). Stoev and Taqqu [42] obtained lower and upper bounds on Hölder exponents which are similar to LFSM results: for all t ∈ R + , H(t) − 1/α ≤ α X,t ≤ H(t) and α X,t = H(t) − 1/α almost surely. Ayache and Hamonier [4] recently investigated the existence of optimal local modulus of continuity.
Theorem 2 can be generalized to the LMSM in the continuous case. More precisely, we assume that the Hurst function satisfies the following assumption, Since the LMSM is clearly a non-homogeneous process, it is natural to focus on the study of the spectrum of singularities localized at t ∈ R + , i.e.
The next statement correspond to an adaptation of Theorem 2 to the LMSM.

Theorem 3. Let X be a linear multifractional stable motion parametrized by α ∈ (1, 2)
and an (H 0 )-Hurst function H. It satisfies almost surely for all t ∈ R and for all σ ∈ is empty for any σ > H(t) − 1 α and ρ > 0 sufficiently small. Remark 6. Theorem 3 extends results presented in [42,43]. In particular, it ensures that the localized multifractal spectrum is equal to Moreover, we observe that Proposition 2 and Theorem 3 still hold when the Hurst function H(·) is a continuous random process. Thereby, similarly to the works of Barral et al. [10] and Durand [17], it provides a class stochastic processes whose spectrum of singularities, given by equation (1.16), is non-homogeneous and random.

Regularity of Lévy processes
In this section, X will designate a Lévy process parametrized by the generating triplet (a, Q, π). Lévy-Itō decomposition states that it can represented as the sum of three independent processes B, N and Y , where B is a d-dimensional Brownian motion, N is a compound Poisson process with drift and Y is a Lévy process characterized by 0, 0, π(dx)1 { x ≤1} . Without any loss of generality, we restrict the study to the time interval [0, 1]. As noticed in [22], the component N does not affect the regularity of X since its trajectories are piecewise linear with a finite number of jumps. Sample path properties of Brownian motion are wellknown and therefore, we first focus in the sequel on the study of the jump process Y .
We know there exists a Poisson measure J(dt, dx) of intensity L 1 ⊗ π such that Y is given by Moreover, as presented in [40] (Theorem 19.2), the convergence is almost surely uniform on any bounded interval. For any m ∈ R + , Y m will denote the following Lévy process x π(dx) . (2.1)

Pointwise exponent
We extend in this section the multifractal spectrum (1.8) to any Lévy process. To begin with, we prove two technical lemmas that will be extensively used in the sequel.
Proof. We note that for any m ∈ R + and all δ > β, Therefore, the stationarity of Lévy processes and Lemma 2.1 yield Let us recall the definition of the collection of random sets (A δ ) δ>0 introduced in [22].
Then, the random set A δ is defined by A δ = lim sup ε→0 + A ε δ . As noticed in [22], if t ∈ A δ , we necessarily have α Y,t ≤ 1 δ . The other side inequality is obtained in the next statement, extending the proof of Proposition 2 from [22].
Proof. Using Lemma 2.2 and Borel-Cantelli lemma, we know that almost surely, there exists M (ω) such that for any m ∈ N ≥ M (ω), Since a linear component does not contribute to the pointwise exponent, we only have to consider the remaining part of the Lévy process to characterize the regularity.
Let u ∈ [0, 1] and m ∈ N such that 2 −m−1 ≤ |t − u| < 2 −m . m can be supposed large enough to satisfy m ≥ M (ω). Let ε 1 = 2 −m/δ . For any jump ∆Y s whose norm is in the is linear in the neighbourhood of t, and therefore can be ignored. For the second integral, we similarly observe that

This last inequality and Equation
Proposition 1 ensures that almost surely Furthermore, since the estimate of the Hausdorff dimension obtained in [22] does not rely on Assumption (1.7), Y satisfies almost surely

2-microlocal frontier: proof of Theorem 1
Let us now study the 2-microlocal frontier of Lévy processes. According to Theorem 3.13 in [32], for all t ∈ [0, 1] and any s > −α Y,t , the sample path Y•(ω) almost surely belongs to the 2-microlocal space C 0,s t . Furthermore, owing to the density of the set of jumps S(ω) in [0, 1], we necessarily have Y•(ω) / ∈ C σ,s t for any σ > 0 and s ∈ R. Hence, since the 2-microlocal frontier is a concave function with left-and right-derivatives between 0 and 1, we obtain that almost surely, for all t ∈ [0, 1] Therefore, we consider in the sequel on the negative component of the 2-microlocal frontier of Y . As outlined in the introduction and according to Definition 2 of 2-microlocal spaces, we need to study the increments around t of the integral of the process Y , i.e.
where b < t is fixed. The form of the polynomial P t depends on β. If β ≥ 1, we only need to remove a linear component equal to u b Y t ds, and therefore the increment simply corresponds to On the other hand, if β < 1, the proof of Proposition 1 induces that we also need to subtract the compensation term s → (s − t) D(0,1) x π(dx). Then, in this case, we have to study For sake of readability, we divide the proof of Theorem 1 and its corollary in different technical lemmas. We begin by obtaining a global upper bound of the frontier.

Lemma 2.3. Almost surely for all
Proof. Let m ∈ N, ε > 0, α = β(1 − 2ε) and γ = β(1 + 4ε). As stated in [22], to prove that α Y,t ≤ 1/β, it is sufficient to exhibit for any ε > 0 a sequence (t n ) n∈N such that ∆Y tn = 2 −n and |t − t n | ≤ 2 −nα . In order to extend this inequality to the 2-microlocal frontier and obtain equation (2.5), we need to reinforce the previous statement and show that in the neighbourhood of t n , there is no other jump of similar size.
More precisely, let consider 2 mα consecutive intervals I j of size 2 −mα . The family (I j ) j forms a cover of [0, 1]. Each I j can be divided into at least 2 m(γ−α) disjoint of intervals I i,j of size 2 −mγ . Finally, let I i,j = I 1 i,j ∪ I 2 i,j ∪ I 3 i,j be the three consecutive intervals of same size inside I i,j . We investigate the probability of the following event Since J is a Poisson measure, A i,j corresponds to the intersection of independent events and we have As described in [12], β can also be defined by β = inf δ ≥ 0 : lim sup r→0 r δ π B(r, 1) < ∞ . Therefore, there exists r 0 > 0 such that for all r ∈ (0, r 0 ], π B(r, 1) ≤ r −β(1+ε) . Then, for all m ∈ N large enough, we obtain According to the definition of β, there also exists an increasing sequence (m n ) n∈N such that for all n ∈ N, π(D(2 −mn , 1)) ≥ 2 mnβ (1−ε) . Therefore, along this sub-sequence, we have P(A i,j ) ≥ 2 −5mnβε−2 · (1 + o n (1)) for any n ∈ N.
Let now consider the event B n,j defined by Since the events (A i,j ) i are identical and independent, we obtain where C is a positive constant. Hence, using Borel-Cantelli lemma, there exists an event Ω 0 of probability 1 such that for any ω ∈ Ω 0 , there exists N (ω) and for all n ≥ N (ω), Let now t ∈ [0, 1], ω ∈ Ω 0 and n ≥ N (ω). There exist i, j ∈ N such that t ∈ I j and ω ∈ A i,j .
Hence, according to the definition of the event A i,j , there is t n ∈ I i,j such that ∆Y tn ≥ 2 −mn and |t − t n | ≤ 2 −mnα . Furthermore, there is no other jump of size greater than 2 −mn(1+ε) in the ball B(t n , ρ n ), where ρ n = 2 −mnγ /3.
Without any loss of generality, we assume in the sequel that Y tn − Y t ≥ 2 −mn−1 is satisfied. As previously outlined, we need to study the increments described in equation (2.4). Specifically, we observe on the interval [t n , t n + ρ n ] that Let obtain an upper bound for the second term. There is no jump of size greater than 2 −mn(1+ε) in the interval (t n , t n + ρ n ]. Therefore, for all s ∈ (t n , t n + ρ n ], we have 1) xπ(dx). Lemma 2.2 provides an upper bound for the first term. Indeed, let m n = m n (1 + ε)δ, where δ = β(1 + ε) > β. Using Borel-Cantelli lemma, we know that for any n ∈ N sufficiently large, for all u, v . Moreover, we observe that |s − t n | ≤ ρ n = 1 3 2 −mnγ ≤ 2 −mnβ(1+ε) 2 = 2 − mnδ . Therefore, using the previous inequalities, we obtain that almost surely for all n ≥ N (ω) To investigate the second integral term, we have distinguish the two different cases introduced above.
The norm of the previous expression is bounded above by as ρ n = 2 −mnγ . 2. If β < 1, we know that the drift u D(0,1) x π(dx) can be removed from the Lévy process.
Therefore, we only have to consider the following quantity x π(dx).
Since it can be assumed that (1 + ε)β ≤ 1, we similarly get where C and C are positive constants independent of n. Finally, since |t − t n | ≤ 2 −mnα , Hence, according to Definition 2 of the 2-microlocal spaces, this last equation (2.9) ensures that Since the inequality holds for any ε ∈ Q > 0 and λ → ε→0 + 1 + 1 β , we obtain the expected result.
As previously, we shall assume that Y t − Y tn ≥ 2 −mn−1 and investigate the increment tn+ρn tn (Y s − Y tn ) ds, where ρ n = 2 −mnα . As stated in Lemma 2.3, for all n ≥ N (ω), Furthermore, the remaining integral also satisfies the following inequality.
To conclude this proof, let us consider the case t ∈ S(ω). We observe that for all u ≥ t, For this last technical lemma, we focus on the particular case α Y,t ∈ (1/2β, 1/β).

Lemma 2.6. Almost surely, for all
Furthermore, for any t ∈ E h , we have σ Y,t (s ) ≤ h+s 2βh for all s ∈ R.
Let now consider h ∈ (1/2β, 1/β) and . Furthermore, as t ∈ E h , there exist sequences (m n ) n∈N and (t n ) n∈N such that ∀n ∈ N, t n ∈ B(t, 2 −mnα ) and ∆Y tn = 2 −mn . Without any loss of generality, we can assume that m n ≥ M for all n ∈ N. Then, since t / ∈ S α mn , we know that for all n ∈ N, there is no jump of size larger than 2 −mn(1+ε) in B(t n , 2 −mnα ).
Before finally proving Theorem 1 and its corollary on the 2-microlocal frontier of Lévy processes, we recall the following result on the increments of a Brownian motion. The proof can be found in [1] (inequality (8.8.26)).

Lemma 2.7. Let B be a d-dimensional Brownian motion. There exists an event Ω 0 of probability one such that for all
Proof of Theorem 1. We use the notations introduced at the beginning of the section. As previously said, the compound Poisson process N can be ignored since it does not influence the final regularity.
Otherwise, the Lévy process X corresponds to the sum of the Brownian motion B and the jump component Y . Still using Lemmas 2.3, 2.5 and 2.6, it is sufficient to prove that almost surely for all t ∈ [0, 1], σ X,t = σ B,t ∧ σ Y,t . Owing to the definition of 2-microlocal frontier, we already know that σ X,t ≥ σ B,t ∧ σ Y,t . Furthermore, when σ B,t (s ) = σ Y,t (s ), it is straight forward to get σ X,t (s ) = σ B,t (s ) ∧ σ Y,t (s ). Therefore, to obtain Theorem 1, we have to prove that almost surely for all t ∈ [0, 1], σ X,t ≤ σ B,t = s → 1/2 + s ∧ 1/2.

1.
If β = β = 2, we only need to slightly modify the proof of Lemma 2.3. More precisely, let consider the same constructed sequence (t n ) n∈N . We observe that since B is almost surely continuous, ∆X tn ≥ 2 −mn and thus, we can still assume that X tn − X t ≥ 2 −mn−1 . Then, we have where we recall that ρ n = 2 −mnβ(1+4ε) /3. This term is negligible in front of ρ n ·(X tn −X t ), and the rest of the proof of Lemma 2.3 ensures that σ X,t (s ) ≤ (1/2 + s ) ∧ 0, for all s ∈ R such that σ X,t (s ) ∈ [−1, 0], which is sufficient to obtain Theorem 1.
Thus, X u − X v ≥ 2 −m(1+2ε)−1 and therefore, for any m sufficiently large, there exists t m ∈ I m,j such that X tm −X t ≥ 2 −m(1+2ε)−2 . Furthermore, t m can be chosen such that there are no jumps of size greater than 2 −m(1+3ε) in B(t m , ρ m ), where ρ n = 2 −mα /3k. Then, using a reasoning similar to the previous point, we obtain for any m ≥ M (ω), This inequality implies that for all s ∈ R such that σ X,t (s ) ∈ [−1, 0], σ X,t (s ) ≤ 1/2+s , which concludes the proof.

Examples of Lévy measures
As previously outlined, the collection of sets ( E h ) h∈R+ considered in Theorem 1 gathers times at which the 2-microlocal regularity is unusual (the slope of the frontier is not equal to 1). In this section, we present examples of Lévy processes which show that for a fixed Blumenthal-Getoor exponent, different situations may occur, depending on the form of the Lévy measure. It is assumed in the sequel that d = 1.

Example 1.
Let π be a Lévy measure such that π((−∞, 0)) = 0. Then, the Lévy process Y with generating triplet (0, 0, π) almost surely satisfies, Proof. According to Theorem 1, we have to prove that for all h ∈ R + , E h = ∅. For that purpose, we extend Lemma 2.5 to any h ∈ [0, 1/β). Let (t n ) n∈N still be a sequence such that |∆Y tn | ≥ 2 −mn and ρ n = t n −t for all n ∈ N. We first assume that β ≥ 1. Similarly to Lemma 2.5, we obtain ( Furthermore, owing to the definition of π, there are only positive jumps in the interval [t, t n + ρ n ]. Hence, if we consider the contribution to the increment Y u − Y t of jumps of size greater than 2 −mn(1+ε) , it is positive and larger than 2 −mn for all u ≥ t n . Therefore, The rest of the proof of Lemma 2.5 holds and ensures the equality. The case β < 1 is treated similarly.
The 2-microlocal frontier obtained in the previous example is proved to be classic at any t ∈ R + . This behaviour is due to the existence of only positive jumps which can not locally compensate each other. Similarly, the proof of Lemma 2.5 focuses on times where the distance between two consecutive jumps of close size is always sufficiently large so that there is no compensation phenomena.
Hence, in order to exhibit some unusual 2-microlocal regularity, we consider in the next example points where jumps are locally compensated at any scale.
Proof. Let β ∈ (0, 1) and α, δ, γ ∈ (β, 2β) such that α < δ < γ. Let the measure π h , where h = 1/δ, be α and δ are supposed to satisfy 2β − 2γ + α > 0. We note that j n → n +∞ since δ > β and 2β − 2γ + α < β. One readily verifies that the Blumenthal-Getoor exponent of π h is equal to β. Moreover, since the measure π h is symmetric, Y is a pure jump process with no linear component. The construction of the example is divided in two parts. We first define a random time set K(ω) and prove it is not empty with a positive probability. Then, we determine the 2-microlocal frontier of points from K(ω) in order to exhibit unusual behaviours.
More precisely, let define an inhomogeneous Galton-Watson process (T n ) n∈N that is used to construct the random set K(ω). Every individual of generation n − 1 represents a an interval I n−1 of size 2 −jn−1δ and the distribution of its offspring is denoted L n .
There exist at least m n = 2 −jn−1δ+jnα−2 intervals ( I i,n ) i of size 3 · 2 −jnα inside I n . Then, for every i, let consider an interval I i,n of size 2 −jnγ centered inside I i,n .
Let p n denotes the probability of the existence of two jumps of size 2 −jn but with different signs inside an interval I i,n and the absence of 2 −jn jumps in the rest of I i,n . It is equal to The distribution of the offspring L n is defined as the number of intervals I i,n which satisfy such a configuration. It follows a Binomial distribution B(m n , p n ) with the following mean, owing to the definition of (j n ) n∈N . Then, to every I n−1 is associated a family of subintervals (I i,n ) i∈Ln such that for all i ∈ L n , I i,n is a subinterval of I i,n of size 2 −jnδ and the distance between I i,n and I i,n is equal to 2 −jnδ .
Let now define the random set K(ω) as following: We It clearly proves that t belongs to A δ for all δ > δ. Since we also know that the distance between t and a jump of size 2 −jn is a at least 2 −jnδ , t / ∈ A δ for all δ < δ. Hence t ∈ E h and the pointwise Hölder exponent α Y,t is equal to h.
Let now study the 2-microlocal frontier of Y at t and set u ∈ B(t, ρ) with ρ sufficiently small. There exists n ∈ N such that 2 −jnδ ≤ |t − u| < 2 −jn−1δ . Two different cases have to be distinguished.

Regularity of linear (multi)fractional stable motion
The linear fractional stable motion (LFSM) is a stochastic process that has been considered by several authors: Maejima [29], Takashima [44], Kôno and Maejima [28], Samorodnitsky and Taqqu [39], Ayache et al. [6], Ayache and Hamonier [4]. Its general integral form is defined by where H ∈ (0, 1), (a + , a − ) ∈ R 2 \ (0, 0) and M α,β is an α-stable random measure on R with Lebesgue control measure λ and skewness intensity β(·) ∈ [−1, 1]. Throughout this paper, it is assumed that β is constant, and equal to zero when α = 1. In this context, for any Borel set A ⊂ R, the characteristic function of M α,β (A) is given by For sake of readability, we consider in the rest of the section the particular case (a + , a − ) = (1, 0) (even though as stated [39], the law of the process depends on values (a + , a − ) chosen).
To begin with, let us obtain in the next statement an alternative representation for the two-parameter field (t, H) → X(t, H) = R (t − u)

Proposition 2.
For all t ∈ R and H ∈ (0, 1), the random variable X(t, H) satisfies 2) where C H = H − 1/α and L is an α-stable Lévy process defined by Proof. Let t ∈ R and H ∈ (0, 1). Since (L t ) t∈R is an α-stable Lévy process, it has càdlàg sample paths. According to [3] (chap. 4.3.4), the theory of the stochastic integration based αstable Lévy processes coincide integrals with respect to α-stable random measure. Therefore, the r.v. X(t, H) is almost surely equal to R (t − u) Let ε > 0 and b < t. Using a classic integration by parts formula, we obtain . Therefore, using equation (3.3) with t = 0 and b < 0, we obtain almost surely When b → −∞, the left-term clearly converges to X(t, H) in L α (Ω). According to [36], we know that almost surely for any ε > 0, lim sup Therefore, as H < 1 and using the dominated convergence theorem, the right-term almost surely converges to the expected integral.
To end this proof, let consider the integral representation in the particular case H = 1/α. In fact, equation (3.2) is a slightly misuse since the expression does not exist. Nevertheless, we prove that it converges almost surely to X(t, 1/α) = L t when H → 1/α. Let first assume that H 1/α and rewrite X(t, H) as The first component of the expression converges to zero since C H → H→1/α 0 and α Y,t a.s. = 1/α. As the second part simply converges to L t , we get the expected limit. The case H 1/α is treated similarly.
Picard [35] determined a similar representation for fractional Brownian motion. We now use this statement to prove Theorem 2.

1.
If H > 1/α, we note that the representation obtained in Proposition 2 is defined almost surely for all t ∈ R. Therefore, let set ω ∈ Ω 0 and t ∈ R. As previously, we can assume that t ∈ [0, 1]. Then, we observe that where b < 0 is fixed. The second term is simply a constant that does not influence the regularity. Similarly, using the dominated convergence theorem, we note that the third one is a C ∞ function on the interval [0, 1], and therefore has no impact on the 2-microlocal frontier. Finally, the first term corresponds to a fractional integral of order H −1/α of the process L. According to the properties satisfied by the 2-microlocal spaces (see e.g. Theorem 1.1 in [24]), we know that almost surely for all t ∈ R, the 2-microlocal frontier σ X,t of X( • , H) simply corresponds to a translation of L's frontier σ L,t .
Similarly to the previous case H > 1/α, the regularity of X only depends on properties of the component One might recognize a Marchaud fractional derivative (see e.g. [38]). Let us modify this expression to exhibit a more classic form of fractional derivative. For almost all s ∈ [0, 1] and ε > 0, we have According to classic real analysis results, the previous expression is differentiable almost everywhere on the interval [0, 1], and therefore for almost all t ∈ [0, 1]. The last two formulas ensure that X( • , H) ∈ L 1 loc (R) a.s., and thus, X( • , H) is a tempered distribution whose 2-microlocal regularity can be determined as well.
Similarly to the previous case, the term t → t b L u (t−u) H−1/α du is a Riemann-Liouville fractional integral of order H −1/α+1 > 0. Hence, at any t ∈ R, the 2-microlocal frontier of the distribution d dt t b L u (t − u) H−1/α du is equal to σ Y,t + H − 1/α. Since the regularity of the second component locally corresponds to spectrum of L, it does not have any influence. Therefore, almost surely for all t ∈ R, the 2-microlocal frontier σ X,t of X( • , H) corresponds to a translation of L's frontier, i.e. σ X,t = σ L,t + H − 1/α.
In both cases that regularity of X can be deduced from L's 2-microlocal frontier. Hence, using Corollary 1 and since L is an α-stable Lévy process, we obtain the spectrum described in equation (1.13).
Another class of processes similar to the LFSM has been introduced and studied in [11,31,15]. Named fractional Lévy processes, it is defined by where d ∈ (0, 1/2) and L is a Lévy process enjoying Q = 0 (no Brownian component), E[L(1)] = 0 and E[L(1) 2 ] < +∞. Owing to this last assumption on L, LFSMs are not fractional Lévy processes. Nevertheless, their multifractal regularity can be determined as well.
Proposition 3. Let X be a a fractional Lévy process parametrized by d ∈ (0, 1/2). It satisfies almost surely for all σ ∈ d − 1, d , where β designates the Blumenthal-Getoor exponent of the Lévy process. Furthermore, for all s ∈ R, E σ,s is empty if σ > d.
Proof. Marquardt [31] established (Theorem 3.3) a representation of fractional Lévy processes equivalent to Proposition 2. Based on this result, an adaptation of Theorem 2 proof yields equation (3.4).
Similarly to the LFSM, this statement refines regularity results established in [11,31] and proves that the multifractal spectrum of a fractional Lévy process is equal to Let us finally conclude this section with the proof of Theorem 3.